Set algebra - Knowlpedia

A set algebra on a set $X$ is a nonempty collection $\mathcal A \subseteq \mathcal P(X)$ such that if $A\in\mathcal A$ then $X\setminus A\in\mathcal A$ , and if $A,B\in\mathcal A$ then $A\cup B\in\mathcal A$ .

Here $\mathcal P(X)$ is the power set of the set $X$ . Closure under complements and finite unions implies closure under finite intersections and finite set differences. A set algebra is the typical domain for a premeasure, and every sigma-algebra is a set algebra.

Examples

For any $X$ , the full collection $\mathcal P(X)$ is a set algebra.
The finite unions of half-open intervals $[a,b)$ , where $a,b\in\mathbb R\cup\{-\infty,+\infty\}$ , form a set algebra on $\mathbb R$ .
On an infinite set $X$ , the collection of all finite subsets of $X$ together with all cofinite subsets of $X$ (those whose complement is finite) is a set algebra.